Analogy of Bombieri’s number for bounded univalent functions
- Authors: Gordienko V.1, Prokhorov D.1
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Affiliations:
- Saratov State University
- Issue: Vol 38, No 3 (2017)
- Pages: 429-434
- Section: Article
- URL: https://journals.rcsi.science/1995-0802/article/view/199212
- DOI: https://doi.org/10.1134/S1995080217030118
- ID: 199212
Cite item
Abstract
Bombieri’s numbers σmn characterize the behavior of the coefficient body for the class S of all holomorphic and univalent functions f in the unit disk normalized by f(z) = z + a2z2 +.... The number σmn is the limit of ratio for Re(n−an) and Re(m−am) as f tends to the Koebe function K(z) = z(1 − z)−2. In particular, σ23=0. We define analogous numbers σmn(M) for the class S(M) ⊂ S of bounded functions |f(z)|< M, |z| < 1, M >1, with the limit of ratio for Re(pn(M) − an) and Re(pm(M) − am) as f tends to the Pick function PM(z) = MK−1(K(z)/M) = z + Σ n=2∞pn(M)zn. We prove that σ23(M) = −4/M, M > 1.
About the authors
V. Gordienko
Saratov State University
Author for correspondence.
Email: valeriygor@mail.ru
Russian Federation, ul. Astrakhanskaya 83, Saratov, 410012
D. Prokhorov
Saratov State University
Email: valeriygor@mail.ru
Russian Federation, ul. Astrakhanskaya 83, Saratov, 410012